Integrand size = 12, antiderivative size = 122 \[ \int \frac {1}{(a+b \tan (c+d x))^3} \, dx=\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {b \left (3 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:
a*(a^2-3*b^2)*x/(a^2+b^2)^3+b*(3*a^2-b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a ^2+b^2)^3/d-1/2*b/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-2*a*b/(a^2+b^2)^2/d/(a+b* tan(d*x+c))
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {\log (i-\tan (c+d x))}{(-i a+b)^3}+\frac {\log (i+\tan (c+d x))}{(i a+b)^3}+\frac {b \left (\left (6 a^2-2 b^2\right ) \log (a+b \tan (c+d x))-\frac {\left (a^2+b^2\right ) \left (5 a^2+b^2+4 a b \tan (c+d x)\right )}{(a+b \tan (c+d x))^2}\right )}{\left (a^2+b^2\right )^3}}{2 d} \] Input:
Integrate[(a + b*Tan[c + d*x])^(-3),x]
Output:
(Log[I - Tan[c + d*x]]/((-I)*a + b)^3 + Log[I + Tan[c + d*x]]/(I*a + b)^3 + (b*((6*a^2 - 2*b^2)*Log[a + b*Tan[c + d*x]] - ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2))/(a^2 + b^2)^3)/(2*d)
Time = 0.67 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3964, 3042, 4012, 3042, 4014, 3042, 4013}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle \frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {2 a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {2 a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a x \left (a^2-3 b^2\right )}{a^2+b^2}}{a^2+b^2}-\frac {2 a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {b \left (3 a^2-b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a x \left (a^2-3 b^2\right )}{a^2+b^2}}{a^2+b^2}-\frac {2 a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {\frac {\frac {b \left (3 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}+\frac {a x \left (a^2-3 b^2\right )}{a^2+b^2}}{a^2+b^2}-\frac {2 a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
Input:
Int[(a + b*Tan[c + d*x])^(-3),x]
Output:
-1/2*b/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((a*(a^2 - 3*b^2)*x)/(a^2 + b^2) + (b*(3*a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b ^2)*d))/(a^2 + b^2) - (2*a*b)/((a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(a^2 + b^2)
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Time = 0.57 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.15
| method | result | size |
| derivativedivides | \(\frac {-\frac {b}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{3}-3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(140\) |
| default | \(\frac {-\frac {b}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{3}-3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(140\) |
| norman | \(\frac {\frac {\left (a^{2}-3 b^{2}\right ) a^{3} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (a^{2}-3 b^{2}\right ) a x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {-5 a^{2} b^{3}-b^{5}}{2 b^{2} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) d}-\frac {2 a \,b^{2} \tan \left (d x +c \right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) d}+\frac {2 b \left (a^{2}-3 b^{2}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(329\) |
| risch | \(-\frac {x}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}-\frac {6 i a^{2} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 i b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 i a^{2} b c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {2 i b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {2 b^{2} \left (-2 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a b +3 a^{2}\right )}{\left (-i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} d \left (i a +b \right )^{3}}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(403\) |
| parallelrisch | \(-\frac {-2 x \tan \left (d x +c \right )^{2} a^{3} b^{4} d +6 x \tan \left (d x +c \right )^{2} a \,b^{6} d -4 x \tan \left (d x +c \right ) a^{4} b^{3} d +12 x \tan \left (d x +c \right ) a^{2} b^{5} d +b^{7}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{2} b^{5}+6 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{3} b^{4}-2 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a \,b^{6}-12 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b^{4}+4 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{6}+6 b^{5} a^{2}-2 x \,a^{5} b^{2} d +6 x \,a^{3} b^{4} d +3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{2} b^{5}+5 a^{4} b^{3}+3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{4} b^{3}-\ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{2} b^{5}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{4} b^{3}+2 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2} b^{5}-\ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} b^{7}+4 \tan \left (d x +c \right ) a^{3} b^{4}+4 \tan \left (d x +c \right ) a \,b^{6}+2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} b^{7}}{2 \left (a +b \tan \left (d x +c \right )\right )^{2} \left (a^{2}+b^{2}\right ) b^{2} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) d}\) | \(453\) |
Input:
int(1/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2*b/(a^2+b^2)/(a+b*tan(d*x+c))^2+b*(3*a^2-b^2)/(a^2+b^2)^3*ln(a+b* tan(d*x+c))-2*a*b/(a^2+b^2)^2/(a+b*tan(d*x+c))+1/(a^2+b^2)^3*(1/2*(-3*a^2* b+b^3)*ln(1+tan(d*x+c)^2)+(a^3-3*a*b^2)*arctan(tan(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (120) = 240\).
Time = 0.09 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.63 \[ \int \frac {1}{(a+b \tan (c+d x))^3} \, dx=-\frac {7 \, a^{2} b^{3} + b^{5} - 2 \, {\left (a^{5} - 3 \, a^{3} b^{2}\right )} d x - {\left (5 \, a^{2} b^{3} - b^{5} + 2 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left (3 \, a^{4} b - a^{2} b^{3} + {\left (3 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (3 \, a^{3} b^{2} - 3 \, a b^{4} + 2 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} d\right )}} \] Input:
integrate(1/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
-1/2*(7*a^2*b^3 + b^5 - 2*(a^5 - 3*a^3*b^2)*d*x - (5*a^2*b^3 - b^5 + 2*(a^ 3*b^2 - 3*a*b^4)*d*x)*tan(d*x + c)^2 - (3*a^4*b - a^2*b^3 + (3*a^2*b^3 - b ^5)*tan(d*x + c)^2 + 2*(3*a^3*b^2 - a*b^4)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - 2*(3*a^3*b^2 - 3*a*b^4 + 2*(a^4*b - 3*a^2*b^3)*d*x)*tan(d*x + c))/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*d*tan(d*x + c)^2 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a* b^7)*d*tan(d*x + c) + (a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d)
Exception generated. \[ \int \frac {1}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate(1/(a+b*tan(d*x+c))**3,x)
Output:
Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr imitive'
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (120) = 240\).
Time = 0.12 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.03 \[ \int \frac {1}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, a b^{2} \tan \left (d x + c\right ) + 5 \, a^{2} b + b^{3}}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )}}{2 \, d} \] Input:
integrate(1/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*(2*(a^3 - 3*a*b^2)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*( 3*a^2*b - b^3)*log(b*tan(d*x + c) + a)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (4*a*b^2*tan(d*x + c) + 5*a^2*b + b^3)/(a^6 + 2*a^4*b^2 + a^2*b^4 + (a^4*b^2 + 2*a^2*b^4 + b^6)*tan(d*x + c)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5 )*tan(d*x + c)))/d
Time = 0.19 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.87 \[ \int \frac {1}{(a+b \tan (c+d x))^3} \, dx=\frac {{\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} + \frac {{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b d + 3 \, a^{4} b^{3} d + 3 \, a^{2} b^{5} d + b^{7} d} - \frac {5 \, a^{4} b + 6 \, a^{2} b^{3} + b^{5} + 4 \, {\left (a^{3} b^{2} + a b^{4}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} d} \] Input:
integrate(1/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
(a^3 - 3*a*b^2)*(d*x + c)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4*d + b^6*d) - 1/ 2*(3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4 *d + b^6*d) + (3*a^2*b^2 - b^4)*log(abs(b*tan(d*x + c) + a))/(a^6*b*d + 3* a^4*b^3*d + 3*a^2*b^5*d + b^7*d) - 1/2*(5*a^4*b + 6*a^2*b^3 + b^5 + 4*(a^3 *b^2 + a*b^4)*tan(d*x + c))/((a^2 + b^2)^3*(b*tan(d*x + c) + a)^2*d)
Time = 1.18 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(a+b \tan (c+d x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {5\,a^2\,b+b^3}{2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {2\,a\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^2-b^2\right )}{d\,{\left (a^2+b^2\right )}^3}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \] Input:
int(1/(a + b*tan(c + d*x))^3,x)
Output:
log(tan(c + d*x) + 1i)/(2*d*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) + (log(ta n(c + d*x) - 1i)*1i)/(2*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) - ((5*a^2*b + b^3)/(2*(a^4 + b^4 + 2*a^2*b^2)) + (2*a*b^2*tan(c + d*x))/(a^4 + b^4 + 2*a^2*b^2))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*tan(c + d*x))) + (b*log(a + b*tan(c + d*x))*(3*a^2 - b^2))/(d*(a^2 + b^2)^3)
Time = 0.23 (sec) , antiderivative size = 543, normalized size of antiderivative = 4.45 \[ \int \frac {1}{(a+b \tan (c+d x))^3} \, dx=\frac {-3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a^{2} b^{3}+\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} b^{5}-6 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{3} b^{2}+2 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a \,b^{4}-3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{4} b +\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{2} b^{3}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a^{2} b^{3}-2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} b^{5}+12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{3} b^{2}-4 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a \,b^{4}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{4} b -2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{2} b^{3}+2 \tan \left (d x +c \right )^{2} a^{3} b^{2} d x +2 \tan \left (d x +c \right )^{2} a^{2} b^{3}-6 \tan \left (d x +c \right )^{2} a \,b^{4} d x +2 \tan \left (d x +c \right )^{2} b^{5}+4 \tan \left (d x +c \right ) a^{4} b d x -12 \tan \left (d x +c \right ) a^{2} b^{3} d x +2 a^{5} d x -3 a^{4} b -6 a^{3} b^{2} d x -4 a^{2} b^{3}-b^{5}}{2 d \left (\tan \left (d x +c \right )^{2} a^{6} b^{2}+3 \tan \left (d x +c \right )^{2} a^{4} b^{4}+3 \tan \left (d x +c \right )^{2} a^{2} b^{6}+\tan \left (d x +c \right )^{2} b^{8}+2 \tan \left (d x +c \right ) a^{7} b +6 \tan \left (d x +c \right ) a^{5} b^{3}+6 \tan \left (d x +c \right ) a^{3} b^{5}+2 \tan \left (d x +c \right ) a \,b^{7}+a^{8}+3 a^{6} b^{2}+3 a^{4} b^{4}+a^{2} b^{6}\right )} \] Input:
int(1/(a+b*tan(d*x+c))^3,x)
Output:
( - 3*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a**2*b**3 + log(tan(c + d*x )**2 + 1)*tan(c + d*x)**2*b**5 - 6*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a **3*b**2 + 2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a*b**4 - 3*log(tan(c + d*x)**2 + 1)*a**4*b + log(tan(c + d*x)**2 + 1)*a**2*b**3 + 6*log(tan(c + d *x)*b + a)*tan(c + d*x)**2*a**2*b**3 - 2*log(tan(c + d*x)*b + a)*tan(c + d *x)**2*b**5 + 12*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**3*b**2 - 4*log(ta n(c + d*x)*b + a)*tan(c + d*x)*a*b**4 + 6*log(tan(c + d*x)*b + a)*a**4*b - 2*log(tan(c + d*x)*b + a)*a**2*b**3 + 2*tan(c + d*x)**2*a**3*b**2*d*x + 2 *tan(c + d*x)**2*a**2*b**3 - 6*tan(c + d*x)**2*a*b**4*d*x + 2*tan(c + d*x) **2*b**5 + 4*tan(c + d*x)*a**4*b*d*x - 12*tan(c + d*x)*a**2*b**3*d*x + 2*a **5*d*x - 3*a**4*b - 6*a**3*b**2*d*x - 4*a**2*b**3 - b**5)/(2*d*(tan(c + d *x)**2*a**6*b**2 + 3*tan(c + d*x)**2*a**4*b**4 + 3*tan(c + d*x)**2*a**2*b* *6 + tan(c + d*x)**2*b**8 + 2*tan(c + d*x)*a**7*b + 6*tan(c + d*x)*a**5*b* *3 + 6*tan(c + d*x)*a**3*b**5 + 2*tan(c + d*x)*a*b**7 + a**8 + 3*a**6*b**2 + 3*a**4*b**4 + a**2*b**6))